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creases about 45% for three of the four
cases and for the A6 quality, the stress
increase is still about 20%.
Runout deviations. In the simulations
performed for this paper,
AGMA radial runout was converted
to tangential spacing errors using the
tangent of the pressure angle. The
runout was considered to be sinusoidal
and a sinusoidal train of spacing errors
was simulated. Values are not presented,
but values were roughly equivalent to
those of individual spacing errors of the
“worst- case spacing” of the sinusoidal
errors.
In converting radial runout to
tangential spacing error, the following
equation was used for each gear:
SE = f idT
2 tan ( t
) 360
N T
where,
SE
peak effective spacing error
FidT radial composite tolerance
t
NT
transverse pressure angle
number of teeth
(3)
Summary
An analysis procedure has
been presented that ac counts for
manufacturing accuracies in evaluating
contact stresses, root stresses and load
distribution factors. The same procedure
may be used for other design metrics
such as film thickness, flash temper ature
and transmission error. The increases in
stresses due to profile and lead deviations
are cer tainly significant, being as high
as 26% for the root stresses of the
example quality A8 helical pinion. The
load distribution factor increased 34%
for the same pinion. Spacing variations
provided even higher stress variation,
with root stresses increas ing by as much
as 45% for the A8 spacing error. Although
not shown, contact stresses and
the load distribution factor also increase
significantly when spacing errors are
applied. Also presented in this paper
is a procedure for selecting appropriate
modi fications that compensate for
misalignment and avoid severe corner
contact and tip interference. General
Spacing and Runout Deviations
Both of these deviations fall in the
domain of the AGMA quality system,
and so are evaluated using the load
distribution analysis. A scheme for
evaluating these deviations on both
static and dynamic stresses has been
performed (Ref. 15), and this method
could in fact be superimposed upon
the methodolo gy described in this
paper. Here, we shall only per form the
static analysis. It is expected that one
can simply superimpose the spacing
and runout devi ation effects with the
lead and profile deviations, without
introducing much error. So for now, we
will only look at the spacing and runout
deviation effects as separate cases.
Spacing Deviations. Here, we shall
evaluate only the worst case tooth-totooth
spacing deviation and will present
results only for root stresses. There
seem to be two possibilities for creating
a worst-case scenario—the first being
when only one tooth is mispositioned,
and the second when many subse quent
teeth are mispositioned by the deviation
toler ance. Figure 17 shows the first case
and Figure 18 shows the second case.
In each case the position of the first
tooth with an error may be positive
(comes into contact early) or negative
(comes into contact late). The negative
condition is shown in the two figures.
These effects are simulated by
essentially shifting the profiles either
forward or backward, depending upon
the sign of the deviation. For positive
devi ations, the tooth with the error
comes into contact early and carries a
disproportionate share of the load. The
values of errors used are the sum of the
square of the spacing deviations of the
pinion and the gear, respectively.
Figure 19 shows the plots of root
stresses for five locations across the
face width (similar to Figure 3) for a
modified gear tooth without spacing
errors. Figure 20 shows the stresses
after the addition of the spacing error,
and one notes that the stresses at the
edge of the tooth increase significantly.
Table 5 summarizes the stress values
for positive and neg ative errors for each
case for both A6 and A8 accu racies.
For quality A8 gears, the root stress inconclusions
are:
• Spacing E V deviations: E N They T have S a
large effect on contact and root
stresses, mainly due to transverse
load sharing (up to 50%).
• Runout has a relatively small
effect, but lower quality gears
still may cause an increase in
stresses.
• Microgeometry changes (profile,
lead and misalignment deviations)
significantly affect
contact stresses (5–10%) and
root stresses (10–25%).
• It is important to start a
design with reasonable
profile and lead modifications,
since this reduces the variability
in stresses due to inaccuracies.
• The best modifications do not
totally eliminate corner contact
or tip interference, but do reduce
their effect. If one totally
eliminates tip interference,
stresses will be higher than
those for the“best” modifications
shown here.
• Although the increases in
stress values due to manufacturing
variability appear to be
quite large relative to current
rating practice, one could justify
these effects as being part of
the uncertainty that is part
of the design factor of safety
(difference between design
allowable stresses and material
property stresses) in the current
rating practice.
Even though many manufacturing
variables have been considered in this
study, there are still numer ous factors that
are affected by manufacturing accuracy
that will influence stress values. Several,
such as tooth thickness, center distance
variation, outside diameter variation and
surface finish varia tion, have not been
included and certainly have possibilities
for future studies. Fortunately, many
designers run min/max calculations for
these parameters so they do in a way get
considered in their evaluations.
Acknowledgements
The author would like to thank the
many sponsors of the Gear and Power
continued
www.geartechnology.com July 2009 GEARTECHNOLOGY 00 59